The present invention generally pertains to estimating the angle of arrival (AOA) of a received signal and is particularly directed to estimating the AOA of a signal received by an array of commutated antenna elements.
Arrays of antenna elements are commonly used for estimating the AOA of a received signal. For tactical signal-intercept applications it is desirable for the signal-intercept hardware to be of minimal size, weight, and power (SWAP). To realize minimal SWAP it is desirable to use a single tuner and to commutate the antenna elements of the array. However, because there is a temporal boundary at the instant of commutation between antenna elements, commutation of the antenna elements results in problems that do not occur with a non-commutated array.
A sampled received signal is a complex signal having a modulus (amplitude) and an argument (angle). Consider a small interval around the boundary of commutation between two antenna elements of an array of antenna elements, and suppose that this interval is much smaller than the reciprocal of the bandwidth of the received signal. Bandwidth determines rate of change. Therefore in close proximity to the boundary the received signal is approximately constant. At the boundary of commutation there is a change related to the AOA of the sampled complex received signal. Theoretically, the AOA affects both the modulus and the argument of a sample of a complex signal received by a low-band antenna, but only the argument of a complex signal received by mid and high-band antennas. At an instant in time that is before or after the boundary by an interval that is on the order of the reciprocal of the bandwidth of the received signal, the received signal itself may change as a result of modulation, and thereby affect the complex signal samples of the received signal.
The AOA may be derived from a received signal sampled immediately adjacent to the boundary (i.e. before and after). The sampled received signal is degraded by noise that is both inside and outside the bandwidth of the received signal; whereby optimal noise performance necessitates rejecting the noise that is outside the bandwidth of the received signal. Ordinarily, such rejection may be accomplished with a simple low-pass filter, provided that the frequency offset is minimal. However, any filter has a transient and this transient will be exhibited at each boundary of commutation. Superficially, it may seem that one need only move away from the boundary to avoid this transient; but moving the sampling of the received signal away from the boundary by the reciprocal of the bandwidth of the received signal permits the modulation aspect of the received signal to affect the observed change in the sampled received signal across the boundary. Thus it is not preferable to merely filter per se. One method that has been used is to filter the sampled received signal with a low-pass filter having a bandwidth larger than the received signal. This approach does not attain optimal noise performance as it does not suppress some noise that is outside the bandwidth of the received signal.
There is also a further problem. At the boundary of commutation there is an interval of time during which electrical switching between the antenna elements occurs, whereupon the resulting samples of the received signal during this interval are unusable. A common method for addressing this further problem is to zero the samples of the received signal observed during this switching interval. This method is incorrect and results in distortion.
In statistical signal processing the problem of estimating the AOA is a parameter estimation problem. As a simple example of a parameter estimation problem, suppose one is given x1=A+v1 and x2=A+v2 where v1 and v2 are random variables, and the objective is to estimate A from x1 and x2. Furthermore, assume that v1 and v2 are independent and identically distributed (i.i.d.) and zero mean. Consider now two different estimates of A, α=x1 and
  β  =                              ϰ          1                +                  ϰ          2                    2        .  The expected value of both of these estimates is A, but the variance of α is twice as large as the variance of β. In other words when there is no noise both of these estimates will be correct, but as noise is introduced β is more likely to be closer to A. It seems that all too often it is the case that an estimate is chosen by manipulating equations that apply in the absence of noise. In the example, this might go something like this. Without noise v1=0 and v2=0, and therefore x1=A and x2=A. Let's use α=x1 as x1=A. This works when there is no noise but
  β  =                    ϰ        1            +              ϰ        2              2  is the superior choice.
Estimation of the AOA is complicated by the fact that it is not possible to estimate the AOA without also estimating the received signal. In other words, there are multiple parameters that must be simultaneously estimated. Many received signals have bandwidths that exceed 100 kHz; and the maximum commutation rate is less than 100 k commutations/second. Thus, commutation boundaries may be separated by more than the reciprocal of the bandwidth of the received signal. With a succession of boundaries the sample of the received signal changes not only with the AOA but also with the modulation of the target signal portion of the received signal.
One method of AOA estimation that has been used previously is to compute the ratio of the observed received signal samples on each side of a boundary. FM discrimination across the boundary is a simple and equivalent alternative method for obtaining the same result. The intuitive appeal of this alternative method is that the result is independent of the modulation of the target signal. (The observed samples of the received signal before and after the boundary are assumed to be the same, whereby they cancel each other in the ratio.) Unfortunately, this alternative method falls into the category of computing the estimate without considering performance in noise, and this alternative method is without rigorous statistical foundation. Additionally, this alternative method also succumbs to problems of suboptimal noise performance and distortion.